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Dynamics and abstract computability: computing invariant measures

Abstract : We consider the question of computing invariant measures from an abstract point of view. Here, computing a measure means finding an algorithm which can output descriptions of the measure up to any precision. We work in a general framework (computable metric spaces) where this problem can be posed precisely. We will find invariant measures as fixed points of the transfer operator. In this case, a general result ensures the computability of isolated fixed points of a computable map. We give general conditions under which the transfer operator is computable on a suitable set. This implies the computability of many “regular enough” invariant measures and among them many physical measures. On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two examples of computable dynamics, one having a physical measure which is not computable and one for which no invariant measure is computable, showing some subtlety in this kind of problems.
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Submitted on : Tuesday, September 14, 2010 - 1:59:49 PM
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Stefano Galatolo, Mathieu Hoyrup, Cristobal Rojas. Dynamics and abstract computability: computing invariant measures. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2011, 29 (1), pp.193-212. ⟨10.3934/dcds.2011.29.193⟩. ⟨inria-00517367⟩

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